Connectivity properties of Julia sets of Weierstrass elliptic functions

l

-d showset oflattice

lished[9,10]

Topology and its Applications 152 (2005) 107–137

www.elsevier.com/locate/topo

Connectivity properties of Julia setsof Weierstrass elliptic functions

Jane Hawkinsa,∗, Lorelei Kossb

a Department of Mathematics, University of North Carolina at Chapel Hill, CB #3250,Chapel Hill, NC 27599-3250, USA

b Department of Mathematics and Computer Science, Dickinson College, P.O. Box 1773,Carlisle, PA 17013, USA

Received 24 September 2003; received in revised form 17 April 2004

Abstract

We discuss the connectivity properties of Julia sets of Weierstrass elliptic℘ functions, accompanied by examples. We give sufficient conditions under which the Julia set is connected anthat triangular lattices satisfy this condition. We also give conditions under which the Fatou℘ contains a toral band and provide an example of an order two elliptic function on a squarewhose Julia set is a Cantor set. 2004 Elsevier B.V. All rights reserved.

MSC:37F10; 30D99; 37F45

Keywords:Complex dynamics; Meromorphic functions; Julia sets

1. Introduction

In this paper we discuss the connectivity of Julia sets of Weierstrass elliptic℘ functions.In [9,10] we studied dynamical properties of Weierstrass elliptic functions and estabresults about the dependence of the dynamics on the underlying lattice. The work in

* Corresponding author.E-mail addresses:[email protected] (J. Hawkins), [email protected] (L. Koss).

0166-8641/$ – see front matter 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.topol.2004.08.018

108 J. Hawkins, L. Koss / Topology and its Applications 152 (2005) 107–137

,8,12,

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s andder isein. The

enttouining asamenents.l bandsve aeding

ice

lnive an

on withet. Wery axeswrittenmakeis also

exate

built on earlier papers on the dynamics of meromorphic functions such as [1–3,5,617] and elliptic functions in [13] and other related papers by these authors.

The existence of examples of Weierstrass elliptic℘ functions with connected Julia seis shown by giving explicit latticesΛ for whichJ (℘Λ) = C∞, the Riemann sphere [9]. Ithe same paper we also give some conditions on the lattice or, equivalently its invag2 andg3, under whichJ (℘Λ) is connected.

In this paper we give general results on connectivity of the Julia sets accompanexamples. There remain several open questions of interest, in particular a completacterization of the connectivity locus for Weierstrass elliptic℘ functions parametrized iany of several ways studied up to now.

The paper is organized as follows. In Section 2 we give the background definitionresults for studying dynamics of elliptic functions. For details on these topics the reaencouraged to read the earlier papers such as [6,9,10,13] and the references thermain results of the paper appear in Section 3; we prove that if℘Λ is the Weierstrass℘function with period latticeΛ, and each critical value lies in a different Fatou compon(or in the Julia set), thenJ (℘Λ) is connected. If two critical values lie in the same Facomponent then we show the existence of a toral band, a Fatou component contagenerator of the fundamental group on the torus. If all three critical values lie in thecomponent ofF(℘Λ), then we prove that the Julia set has uncountably many compoand, under the additional hypothesis of hyperbolicity, we show thatJ (℘Λ) is a Cantor setWe also give a variety of examples of lattices whose associated Julia sets have torain F(℘Λ). In Section 3.2 we generalize to other hyperbolic elliptic functions and gisufficient condition for the Julia set to be a Cantor set. One corollary of the precresults is that ifΛ is a lattice such that each Fatou component of℘Λ fits entirely in onefundamental region ofΛ, thenJ (℘Λ) is connected. Another is that for any triangular lattΛ, J (℘Λ) is connected.

In Section 4 we give an example of an order two elliptic functionfΛ, which is a rationaexpression of℘Λ for a particular square latticeΛ, such thatJ (f ) is a Cantor set. In Sectio5 we establish additional topological properties of some specialized examples; we gexample of a lattice whose associated Julia set is connected but whose intersectithe real and imaginary axes (and parallel lines a fixed distance apart) is a Cantor salso give some lattices whose associated Julia sets contain the real and imagina(and countably many parallel lines) and have a nonempty Fatou set. The paper iswith color graphics and several color references appear in the text. We attempt tothe reference to the graphics clear enough so that in black and white the meaningapparent.

2. The basics on Weierstrass ℘ dynamics

Let λ1, λ2 ∈ C\{0} such thatλ2/λ1 /∈ R. We define a lattice of points in the complplane byΛ = [λ1, λ2] := {mλ1+nλ2: m,n ∈ Z}. Two different sets of vectors can gener

J. Hawkins, L. Koss / Topology and its Applications 152 (2005) 107–137 109

-

attices

be

the same latticeΛ; if Λ = [λ1, λ2], then all other generatorsλ3, λ4 of Λ are obtained bymultiplying the vector(λ1, λ2) by the matrix

A =(

a b

c d

)

with a, b, c, d ∈ Z andad − bc = 1.We can viewΛ as a group acting onC by translation, eachω ∈ Λ inducing the trans

formation ofC:

Tω : z �→ z + ω.

Definition 2.1. A closed, connected subsetQ of C is defined to be afundamental regionfor Λ if

(1) for eachz ∈ C, Q contains at least one point in the sameΛ-orbit asz;(2) no two points in the interior ofQ are in the sameΛ-orbit.

If Q is any fundamental region forΛ, then for anys ∈ C, the set

Q + s = {z + s: z ∈ Q}is also a fundamental region. If we chooseQ to be a parallelogram we callQ a periodparallelogramfor Λ.

The “appearance” of a latticeΛ = [λ1, λ2] is determined by the ratioτ = λ2/λ1. (Wegenerally choose the generators so that Im(τ ) > 0.) If Λ = [λ1, λ2], and k �= 0 is anycomplex number, thenkΛ is the lattice defined by takingkλ for eachλ ∈ Λ; kΛ is saidto besimilar to Λ. For example, the latticeΛτ = [1, τ ] is similar to the latticeΛ = λ1Λτ .Similarity is an equivalence relation between lattices, and an equivalence class of lis called ashape.

Definition 2.2.

(1) Λ = [λ1, λ2] is real rectangularif there exist generators such thatλ1 is real andλ2 ispurely imaginary. Any lattice similar to a real rectangular lattice isrectangular.

(2) Λ = [λ1, λ2] is real rhombicif there exist generators such thatλ2 = λ1. Any similarlattice isrhombic.

(3) A latticeΛ is squareif iΛ = Λ. (Equivalently,Λ is square if it is similar to a latticegenerated by[λ,λ i], for someλ > 0.)

(4) A latticeΛ is triangular if Λ = e2πi/3Λ in which case a period parallelogram canmade from two equilateral triangles.

In each of cases (1)–(3) the period parallelogram with vertices 0, λ1, λ2, and λ3 :=λ1 + λ2 can be chosen to look rectangular, rhombic, or square, respectively.

Definition 2.3. An elliptic functionis a meromorphic function inC which is periodic withrespect to a latticeΛ.


h is

tion

:

fi-

l

For anyz ∈ C and any latticeΛ, theWeierstrass elliptic functionis defined by

℘Λ(z) = 1

z2+

∑w∈Λ\{0}

(1

(z − w)2− 1

w2

).

Replacing everyz by −z in the definition we see that℘Λ is an even function. The map℘Λ

is meromorphic, periodic with respect toΛ, and has order 2.The derivative of the Weierstrass elliptic function is also an elliptic function whic

periodic with respect toΛ defined by

℘′Λ(z) = −2

∑w∈Λ

1

(z − w)3.

The Weierstrass elliptic function and its derivative are related by the differential equa

℘′Λ(z)2 = 4℘Λ(z)3 − g2℘Λ(z) − g3, (1)

whereg2(Λ) = 60∑

w∈Λ\{0} w−4 andg3(Λ) = 140∑

w∈Λ\{0} w−6.The numbersg2(Λ) andg3(Λ) are invariants of the latticeΛ in the following sense

if g2(Λ) = g2(Λ′) andg3(Λ) = g3(Λ

′), thenΛ = Λ′. Furthermore given anyg2 andg3such thatg3

2 − 27g23 �= 0 there exists a latticeΛ havingg2 = g2(Λ) andg3 = g3(Λ) as its

invariants [7].

We mention the following classical result [7].

Theorem 2.1. Every elliptic functionfΛ with period latticeΛ can be written asfΛ(z) =R(℘Λ(z)) + ℘Λ

′(z)Q(℘Λ(z)), whereR andQ are rational functions with complex coefcients.

Corollary 2.2. Every elliptic function isn-to-one, wheren > 1, within each fundamentaregion. The Weierstrass elliptic℘Λ function is two-to-one in each fundamental region.

Theorem 2.3 [7]. For Λτ = [1, τ ], the functionsgi(τ ) = gi(Λτ ), i = 2,3, are analyticfunctions ofτ in the open upper half planeIm(τ ) > 0.

We have the following homogeneity in the invariantsg2 andg3 [10].

Lemma 2.4. For latticesΛ andΛ′, Λ′ = kΛ ⇐⇒g2(Λ

′) = k−4g2(Λ) and g3(Λ′) = k−6g3(Λ).

Theorem 2.5 [11]. The following are equivalent:

(1) ℘Λ(z) = ℘Λ(z);(2) Λ is a real lattice;(3) g2, g3 ∈ R.


ing

s defi-

ite

nitely

t

-

int.

tions

For any latticeΛ, the Weierstrass elliptic function and its derivative satisfy the followproperties: fork ∈ C\{0},

℘kΛ(ku) = 1

k2℘Λ(u) (homogeneity of℘Λ),

℘′kΛ(ku) = 1

k3℘′

Λ(u) (homogeneity of℘′Λ). (2)

Verification of the homogeneity properties can be seen by substitution into the serienitions.

The critical values of the Weierstrass elliptic function on an arbitrary latticeΛ =[λ1, λ2] are as follows. Forj = 1,2, notice that℘Λ(λj − z) = ℘Λ(z) for all z. Takingderivatives of both sides we obtain−℘′

Λ(λj − z) = ℘′Λ(z). Substitutingz = λ1/2, λ2/2, or

λ3/2, we see that℘′Λ(z) = 0 at these values. We use the notation

e1 = ℘Λ

(λ1

2

), e2 = ℘Λ

(λ2

2

), e3 = ℘Λ

(λ3

2

)

to denote the critical values. Sincee1, e2, e3 are the distinct zeros of Eq. (1), we also wr

℘′Λ(z)2 = 4

(℘Λ(z) − e1

)(℘Λ(z) − e2

)(℘Λ(z) − e3

). (3)

Equating like terms in Eqs. (1) and (3), we obtain

e1 + e2 + e3 = 0, e1e3 + e2e3 + e1e2 = −g2

4, e1e2e3 = g3

4. (4)

In the real lattice case, the homogeneity property, Eq. (2), is used to produce infimany lattices with real superattracting fixed points.

Lemma 2.6. LetΓ be a real lattice such thatγ /2 is the smallest positive real critical poinander is the largest real critical value. Ifm is any odd integer andk = 3

√2er/(mγ ) (taking

the real root) then the latticeΛ = kΓ has a real superattracting fixed point atmkγ/2.

Proof. ClearlyΛ is a real lattice. Eq. (2) implies thatkγ /2 is a real critical point for℘Λ.Sincem is odd, periodicity implies that℘Λ(mkγ/2) = ℘Λ(kγ /2). Further, the homogeneity property implies that

℘Λ

(kγ

2

)= ℘kΓ

(kγ

2

)= 1

k2℘Γ

(γ

2

)= er

k2= mkγ

2. �

Fig. 1 shows the graph of℘Λ on R for a lattice which has a superattracting fixed po

2.1. Fatou and Julia sets for elliptic functions

We review the basic dynamical definitions and properties for meromorphic funcwhich appear in [1,4–6]. Letf :C → C∞ be a meromorphic function whereC∞ =C ∪ {∞} is the Riemann sphere. TheFatou setF(f ) is the set of pointsz ∈ C∞ such that{f n: n ∈ N} is defined and normal in some neighborhood ofz. TheJulia setis the comple-ment of the Fatou set on the sphere,J (f ) = C∞\F(f ). Notice thatC∞\⋃n�0 f −n(∞)


l

re-

of

Fig. 1. Graph of℘Λ|R with a superattracting fixed point.

is the largest open set where all iterates are defined. Sincef (C∞\⋃n�0 f −n(∞)) ⊂C∞\⋃n�0 f −n(∞), Montel’s theorem implies that

J (f ) =⋃n�0

f −n(∞).

Let Crit(f ) denote the set of critical points off , i.e.,

Crit(f ) = {z: f ′(z) = 0

}.

If z0 is a critical point thenf (z0) is acritical value. For each lattice,℘Λ has three criticavalues and no asymptotic values. Thesingular setSing(f ) of f is the set of critical andfinite asymptotic values off and their limit points. A function is calledClassS if f hasonly finitely many critical and asymptotic values; for each latticeΛ, every elliptic functionwith period latticeΛ is of ClassS. Thepostcritical setof ℘Λ is:

P(℘Λ) =⋃n�0

℘nΛ(e1 ∪ e2 ∪ e3).

For a meromorphic functionf , a pointz0 is periodicof periodp if there exists ap � 1such thatf p(z0) = z0. We also call the set{z0, f (z0), . . . , f

p−1(z0)} a p-cycle. Themul-tiplier of a pointz0 of periodp is the derivative(f p)′(z0). A periodic pointz0 is calledattracting, repelling,or neutral if |(f p)′(z0)| is less than, greater than, or equal to 1,spectively. If|(f p)′(z0)| = 0 thenz0 is called asuperattractingperiodic point. As in thecase of rational maps, the Julia set is the closure of the repelling periodic points [1].

SupposeU is a connected component of the Fatou set. We say thatU is preperiodicifthere existsn > m � 0 such thatf n(U) = f m(U), and the minimum ofn − m = p for allsuchn,m is theperiodof the cycle.

Since every elliptic function is of ClassS the basic dynamics are similar to thoserational maps with the exception of the poles. The first result holds for all ClassS functionsas was shown in [4, Theorem 12] and [17].

Theorem 2.7. For any latticeΛ, the Fatou set of an elliptic functionfΛ with period latticeΛ has no wandering domains and no Baker domains.


o allical

tion.

ou

ic

si-to the

ntains

the

In particular, Sullivan’s No Wandering Domains Theorem holds in this setting sFatou components offΛ are preperiodic. Because there are only finitely many critvalues, we have a bound on the number of attracting periodic points that can occur.

The next result was proved in [10]; it is only known for the Weierstrass elliptic func

Theorem 2.8. For any latticeΛ, ℘Λ has no cycle of Herman rings.

We summarize this discussion with the following result.

Theorem 2.9. For any latticeΛ, at most three different types of forward invariant Fatcycles can occur for℘Λ, and each periodic Fatou component contains one of these:

(1) a linearizing neighborhood of an attracting periodic point;(2) a Böttcher neighborhood of a superattracting periodic point;(3) an attracting Leau petal for a periodic parabolic point. The periodic point is inJ (℘Λ);(4) a periodic Siegel disk containing an irrationally neutral periodic point.

The proof of Lemma 2.10 is given for℘Λ in [10] but remains the same for the elliptfunctionfΛ.

Lemma 2.10. If Λ is any lattice andfΛ is an elliptic function with period latticeΛ, then

(1) J (fΛ) + Λ = J (fΛ), and(2) F(fΛ) + Λ = F(fΛ).

The periodicity offΛ andJ (fΛ) with respect to the lattice gives rise to different posbilities for connectivity of a component when the Fatou and Julia sets are projectedtorus. If we write

πΛ :C → C/Λ

for the usual toral quotient map we make the following definition.

Definition 2.4. A Fatou componentFo of the mapfΛ is called atoral bandif Fo containsan open subsetU which is simply connected inC, butπΛ(U) is not simply connected oC/Λ. In other words,U projects down to a topological band around the torus and cona homotopically nontrivial curve.

We also introduce the notion of a double toral band.

Definition 2.5. Suppose that we have an elliptic functionfΛ with period latticeΛ. Ifwe have a componentW ⊂ F(fΛ) which contains a simple closed loop which formsboundary of a fundamental region forΛ, then we sayW is adouble toral band.

Clearly a double toral bandW contains a toral band as is shown by the next result.


o-ental

ly

as

ut this

reare

al be-lts in

nts of

th

Proposition 2.1 [10]. For any latticeΛ, fΛ has a toral band if and only if there is a compnent of the Fatou set which is not completely contained in the interior of one fundamregion ofC/Λ.

We conclude this section with a result about Siegel disks for elliptic functions.

Proposition 2.2. AssumefΛ has a cycle of Siegel disksS = {U1, . . . ,Up}, the unionof pairwise disjoint open simply connected sets, withfΛ(S) = S,fΛ(Uj ) = Uj+1, j =1, . . . , p − 1, fΛ(Up) = U1, andf

pΛ(Uj ) = Uj for j = 1, . . . , p. Then the following hold:

(1) EachUj is completely contained in one fundamental region ofΛ.(2) If z ∈ S, then for everyλ ∈ Λ,λ �= 0, z + λ /∈ S.

If fΛ = ℘Λ, then the following also holds:(3) If c is any critical point in the fundamental region containingUj , for anyj = 1, . . . , p,

and if z ∈ S is of the formz = c + w, thenv = c − w (the point symmetric toz aboutthe critical point) is not inS.

Proof. Suppose there is a simply connected domainUj as above which is not completecontained in one fundamental region forΛ. Then there existsz ∈ Uj andλ ∈ Λ such thatz, z + λ ∈ Uj . Since we have a Siegel disk cycle,f

pΛ |Uj

is injective and is conformallyconjugate to irrational rotation on a disk; butfΛ(z) = fΛ(z + Λ), sof

pΛ(z) = f

pΛ(z + λ),

which cannot occur. Therefore there is no suchz in Uj .If z ∈ Uj , andz + λ ∈ Uk , k �= j , thenfΛ(z) ∈ Uj+1, while fΛ(z + λ) ∈ Uk+1 (both

mod p), which contradicts the disjointness of theU ′j s sincefΛ(z) = fΛ(z + λ).

The last statement follows since℘Λ(c + w) = ℘Λ(c − w); this holds since 2c ∈ Λ, and℘Λ(w + c) = ℘Λ(−w − c) = ℘Λ(−w − c + 2c). The rest of the argument is exactlyabove. �

3. Results on connectivity properties of J(℘Λ)

We now turn to the main connectivity results of this paper. We assume throughosection we are considering the Weierstrass elliptic℘ function ℘Λ with period latticeΛ.First we give sufficient conditions under whichJ (℘Λ) is connected. The proof given heis based on a similar result for polynomials given by Milnor in [15]. Although thereinfinitely many critical points for℘Λ, there are exactly three critical values and℘Λ islocally two-to-one in each fundamental region. Therefore one can expect dynamichavior somewhat similar to that for low degree rational maps. In particular the resuthis section show that the full spectrum of connectivity possible for Fatou componemeromorphic functions, such as that shown in [2], cannot occur in this setting.

Theorem 3.1. If Λ is a lattice such that each critical value of℘Λ that lies in the Fatou seis the only critical value in that component, thenJ (℘Λ) is connected. In particular, if eacFatou component contains either0 or 1 critical value, thenJ (℘Λ) is connected.


ere

h

so byd

e es-ce

is ar

nt,

in

r

ent ofalue

. Inl point,

Proof. If F(℘Λ) = ∅, thenJ (℘Λ) = C∞ is connected. Therefore we assume that this a nonempty Fatou set, and consider a loopγ in any Fatou componentW of ℘Λ. It isenough to show thatγ shrinks to a point inW sinceJ (℘Λ) is connected if and only if eaccomponent ofF(℘Λ) is simply connected (see, e.g., [9, Proposition 6.2(1)]). Since℘Λ hasonly finitely many critical values, by Theorem 2.7 there are no wandering domains,Sullivan’s nonwandering theorem some forward image℘k

Λ(γ ) lies in a simply connecteregionU ⊂ F(℘Λ), andU is in a Fatou component containing one of these:

(1) a linearizing neighborhood of an attracting periodic point;(2) a Böttcher neighborhood of a superattracting periodic point;(3) an attracting Leau petal for a periodic parabolic point;(4) a periodic Siegel disk (i.e., one component of a Siegel disk cycle).

We established in [10] that there are no Herman rings, and in Proposition 2.2 wtablished that any Siegel disk is contained in one fundamental region for the lattiΛ.Without loss of generality, we can assume thatU ⊂ Qo, a single period parallelogram(with appropriate boundaries ofQo chosen not to intersect the boundary ofU ).

Furthermore sinceU is simply connected, we can choose it so that its boundarysimple closed curve not passing through a critical value, andU contains either one ono critical values. Using induction onk, we show that℘−k

Λ (U) consists of only simplyconnected components. Thereforeγ is homotopic to a point. For the inductive argumewe first show that℘−1

Λ (U) has simply connected components.

Case1: If U contains no critical value, then in each fundamental region,℘−1Λ (U) consists

of two disjoint simply connected regions.

Case2: If U contains one critical value, then℘−1Λ (U) is a single simply connected set

each fundamental region, bounded by a simple closed curve, and which maps ontoU by aramified two-fold covering. This follows from the fact that℘Λ is locally two-to-one neaeach critical point and the branched covering℘−1

Λ (U) → U is two-to-one in each region.

The inductive step onk is the same. By adjusting the original choice ofU a little (andat most finitely many times), we can assume that each simply connected compon℘−1

Λ (U) contains no critical value in its boundary and contains either 0 or 1 critical vin it, so we repeat the argument.�Corollary 3.2. If all critical values of℘Λ lie in J (℘Λ), thenJ (℘Λ) is connected.

Proof. Assume all critical values lie inJ (℘Λ). If F(℘Λ) is empty, theJ (℘Λ) = C∞ andwe are done. IfF(℘Λ) �= ∅, then the only possibility is that we have Siegel disk cyclesany case by hypothesis each Fatou component contains no critical value or criticaso we can apply Theorem 3.1.�Corollary 3.3. If Λ is a triangular lattice, thenJ (℘Λ) is connected.

Proof. We showed in [10] that for any triangular lattice the critical valuese1, e2, ande3belong to disjoint components ofF(℘Λ) or they all belong toJ (℘Λ). Moreover we proved


em 3.1

ow ifoth cantoral

es,

that

t

enth in

ral

nt

nda-

t con-more

1, the

that forj = 1,2,3, e2πi/3ej = ej+1 (mod 3) and there is only one “type” of critical valubehavior, which is then rotated to the other two critical values. We then apply Theoreand Corollary 3.2. �

We have defined and given examples of toral Fatou bands in [9]. We do not yet knthe Julia set associated with a toral band is connected, disconnected, or whether boccur. However, we prove the following result which gives sufficient conditions for aband to exist.

Theorem 3.4. If ℘Λ has a Fatou component that contains (at least) two critical valuthen℘Λ has a toral band.

Proof. Writing Λ = [λ1, λ2], we label three critical pointsc1 = λ1/2, c2 = λ2/2, andc3 = λ3/2 = (λ1 + λ2)/2 = c1 + c2; we treat three cases in a similar way. Supposee1 = ℘Λ(c1) and e3 = ℘Λ(c3) both lie in a Fatou componentW . SinceW is open andconnected it is path connected. Letγ be a path frome3 to e1 in W . LetV be the componenof ℘−1

Λ (W) that containsc3. Then one component of℘−1Λ (γ ) connectsc3 to c1 or c1 + λ2

in V because either of these paths is contained in the period parallelogramQ = {sλ1 +tλ2: 0 � s, t � 1}. If ℘−1

Λ (γ ) connectsc3 to c1 we can write this path asc3 + z(t) withz(0) = 0 andz(1) = c1 − c3 = −c2 (or z(0) = 0 andz(1) = c1 + λ2 − c3 = c2). UsingProposition 5.1 in [10], fori = 1,2,3, ci + z ∈ V if and only if ci − z ∈ V ; i.e., we havesymmetry of a Fatou component about any critical point in it. Thereforec3 − z(t) is also apath inV connectingc3 to c1 + λ2. Alternatively,V might containc3 andc1 + λ2.

In this case we repeat the argument above, using the symmetry of the componV

about critical points to continue the path so that, in either case, we end up with a patV

passing throughc3 and connectingc1 to c1 + λ2.Since bothc1 andc1 + λ2 lie in V andV is open, the componentV is not completely

contained in one fundamental region ofC/Λ, so by [10, Proposition 5.2] we have a toFatou band.

If e2 and e3 lie in the same component, then a similar argument shows thatc2 andc2 + λ1 both lie in the same Fatou component.

For the third case, ife1 and e2 lie in the same Fatou componentW , then a similarargument gives a pathγ from e2 to e1 in W , hence a path inV , a component of℘−1

Λ (W),written asc2+z(t) from c2 to c1 or c1−λ1+λ2. The path lies in a single Fatou componeV . Then by symmetry we have thatc2 − z(t) lies in V for all t ∈ [0,1], so in particularc2 − z(1) = −c1 + λ2 = c1 − λ1 + λ2 ∈ V . �Corollary 3.5. If every periodic Fatou component is completely contained in one fumental period ofΛ, thenJ (℘Λ) is connected.

Proof. We claim first that the hypothesis implies that each periodic Fatou componentains at most one critical value. If there is a Fatou component that contains 2 orcritical values, then we apply Theorem 3.4 to obtain a toral band. By Proposition 2.toral band does not fit in one fundamental region, which gives a contradiction.


r

erent

to the

e ex-tionstions

ause

ealuesalnt

ycle

xpointto

tnto a

same

of this

Therefore we have established that every Fatou component for℘Λ contains either 0 o1 critical value and we can apply Theorem 3.1.�3.1. Toral band examples

Suppose exactly two critical values are in one Fatou component of℘Λ; then by Theo-rem 3.4 there is a toral Fatou band, and three possible cases:

(1) the third critical value is in the Julia set,(2) the third critical value lies in a different Fatou component corresponding to a diff

cycle, or,(3) there is exactly one nonrepelling cycle and the third critical value is attracted

same cycle as the other two but lies in a separate component.

The existence of a lattice with a toral band was proved by the authors in [10]. Thamples shown in this section, found numerically using Mathematica, provide illustraof the three possibilities listed above for the critical values of Weierstrass elliptic funcwith toral bands. In the first two of these examples the latticeΛ is real.

Example 3.6. ℘Λ has a toral band and the third critical value lies in the Julia set (becit is a pole).

We considerg2 ≈ 26.5626 andg3 ≈ −26.2672; the corresponding latticeΛ is real rec-tangular (see [7]). For these values℘Λ has an attracting fixed pointp ≈ 1.5566, whichis plotted (in orange) in Fig. 2.1 One period parallelogram forΛ is outlined in light blue.The light points (colored yellow) in Figs. 2 and 3 iterate top and hence belong to thFatou set. The dark points (colored blue) lie in the Julia set. In Fig. 3 the critical ve1 ≈ 1.5539, e2 ≈ −2.9746, ande3 ≈ 1.4206 are plotted (in purple). Two of the criticvalues lie in the component containingp, and the third critical value is the lattice poi−λ1 ≈ −2.9746, a pole, so is in the Julia set.

Example 3.7. ℘Λ has a toral band and the third critical value belongs to a different cof Fatou components than the first two critical values.

In this exampleg2 ≈ 27.85 andg3 ≈ −28.338 soΛ is real rhombic. The two complecritical values lie in the same Fatou component and converge to an attracting fixedat p ≈ 1.542; the medium shaded points (colored tan) in Figs. 4 and 5 convergep

under iteration. Also,℘Λ has a superattracting fixed point atq ≈ −3.047, and the lighpoints (colored yellow) iterate toq. The fixed pointsp andq are marked (in orange) iFig. 4. The dark points (colored navy blue) lie in the Julia set. In Fig. 5 we zoom inneighborhood of the superattracting fixed pointq.

Example 3.8. ℘Λ has a toral band and the third critical value is associated with thecycle of Fatou components as the first two, but lies in a different Fatou component.

1 For interpretation of the references to colour in the figures, the reader is referred to the web versionarticle.


s. 6have

in on

ame

ase.

Fig. 2. The attracting fixed point of the Fatou set of Example 3.6.

Fig. 3. Critical values of Example 3.6.

We setg2 ≈ −1.451−4.984i andg3 ≈ −2.136−0.801i, so the corresponding latticeΛis rhombic. Then℘Λ has an attracting two-cycle at{p1,p2} ≈ {−0.422+0.517i,−0.537+2.25i}, which is plotted (in orange) in Fig. 6. The lightest points (colored yellow) in Figand 7 converge to this cycle. The dark points (navy blue) lie in the Julia set, and weshown the boundary of one period parallelogram (in light blue). In Fig. 7, we zoompart of Fig. 6 and plot the critical valuese1 ≈ 0.790− 1.053i, e2 ≈ −0.423+ 0.517i, ande3 ≈ −0.367+ 0.536i (in purple).

3.2. Disconnected Julia sets

The fourth possibility for a toral band is that all three critical values lie in the sFatou component of℘Λ. Though we do not know if this can occur for any latticeΛ for theWeierstrass elliptic function℘Λ, we show that the Julia set is not connected in this c


touthereficientr two

ry

tly oneg fixed

e

Fig. 4. Fixed points and the Fatou set of Example 3.7.

Fig. 5. Superattracting fixed point of Example 3.7.

If in addition ℘Λ is hyperbolic, we show that having all the critical values in one Facomponent forcesJ (℘Λ) to be a Cantor set. We show that under these conditionsexists a double toral band component (see Definition 2.5), and show this is a sufcondition for a Cantor Julia set. We use this result to give an example of an ordeelliptic function whose Julia set is a Cantor set.

We begin by showing that in order forJ (℘Λ) to be totally disconnected, it is necessathat all critical values lie in the same Fatou component. Recall that iffΛ is an ellipticfunction andJ (fΛ) is totally disconnected, thenJ (fΛ) is called aCantor setsince it ishomeomorphic to the classical middle thirds Cantor set. In this case there is exacFatou component which is therefore completely invariant, so there is one nonrepellinpoint. Meromorphic functions with Cantor Julia sets have been shown to exist in [6].

Throughout this section the notationfΛ refers to any elliptic function with period latticΛ; the notation℘Λ, as usual, refers to the Weierstrass℘ function with period latticeΛ.


allpletely

a to-,ponentonding

Fig. 6. Two cycle and the Fatou set of Example 3.8.

Fig. 7. Critical values of Example 3.8.

If the period lattice is understood or irrelevant, we frequently will usef for an ellipticfunction; all functions in this section are elliptic.

Proposition 3.1. If J (f ) is totally disconnected and contains no critical point, thencritical values lie in the same component of the Fatou set, the component is cominvariant, and it corresponds to a nonrepelling fixed point.

Proof. If J (f ) is totally disconnected, the Fatou set contains only one component;tally disconnected set cannot separate the sphere. SinceJ (f ) contains no critical pointwe cannot have any Siegel disks or Herman rings. Therefore the only Fatou compossible is a Böttcher domain, an attracting domain, or a Leau petal, each correspto a fixed point. �


ntionalcon-, butorphic

et

rd

-

k

We give some results in the other direction. Recall that ifJ (f ) is disconnected theit consists of uncountably many components; the proof of this is the same as for ramaps (e.g., see [16]). We turn to the definition of a hyperbolic elliptic function; thecept of hyperbolicity for a meromorphic function is similar to that for a rational mapthe equivalent notions in the rational settings are not always the same for meromfunctions (e.g., see [17]).

Definition 3.1. We say that an elliptic function ishyperbolicif J (f ) is disjoint fromP(f ).

We adapt the following result from [17, Theorem C] to our setting. We define the s

An(f ) = {z ∈ C: f n is not analytic atz

}.

Sincef is elliptic,

An(f ) =n⋃

j=1

f −j({∞}).

We therefore denote the set of prepoles forf by

A =⋃n�1

An.

We say thatω is anorder k prepoleif ℘kΛ(ω) = ∞; so a pole is an order 1 prepole.

Theorem 3.9 [17]. If an elliptic functionf is hyperbolic then there existK > 1 andc > 0such that

∣∣(f n)′(z)

∣∣ > cKn |f n(z)| + 1

|z| + 1,

for eachz ∈ J (f ) \ An(f ),n ∈ N.

Remark 3.1. For a fixed latticeΛ, and elliptic functionf = fΛ, we have thatf (z + λ) =f (z) andf ′(z+λ) = f ′(z) for everyλ ∈ Λ. Therefore we can prove the following standaresult about hyperbolic functions.

Theorem 3.10. An elliptic functionf is hyperbolic if and only if there existK > 1 andC > 0, C = C(Λ), such that∣∣(f n

)′(z)

∣∣ > CKn, (5)

for eachz ∈ J (f ) \ An(f ),n ∈ N.

Proof. If f is hyperbolic, by Theorem 3.9 there is aK > 1 andc > 0 such that|(f n)′(z)| >cKn |f n(z)|+1

|z|+1 , for eachz ∈ J (f ) \ An(f ),n ∈ N. In particular, forz in one period parallel

ogramQ ⊂ B(0,R) = {z: |z| < R} we have that|(f n)′(z)| > cKn 1R+1, so we just choose

C = c/(R + 1) for z ∈ Q ∩ (J (f ) \ An(f )), for eachn ∈ N. However from the remarabove, we have that the same constantC works for allz ∈ J (f ) \ An(f ), n ∈ N.


n (5)ss 1 ines

touo-

in for

havethe

The converse follows as in the case of rational maps. If the expanding conditioholds, there can be no critical points onJ (f ), soJ (f ) �= C∞ and we have no Siegel diskor Herman rings. There cannot be parabolic cycles (i.e., periodic points of moduluJ (f )) either. Therefore all components ofF(f ) are mapped forward to attracting cyclsoJ (f ) is disjoint fromP(f ). �Corollary 3.11. If f is hyperbolic, then there exists anr > 1, an s ∈ N, and a neighbor-hoodU of J (f ), such that∣∣(f s

)′(z)

∣∣ > r for all z ∈ U \ As(f ).

We state the main result of this section.

Theorem 3.12. If ℘Λ is hyperbolic and all the critical values of℘Λ are contained in oneFatou component, thenJ (℘Λ) is a Cantor set.

Proof. Writing Λ = [λ1, λ2], we label three critical pointsc1 = λ1/2, c2 = λ2/2, andc3 = λ3/2= (λ1 + λ2)/2 = c1 + c2. By hypothesis all three critical values lie in one Facomponent, sayU . Since each fundamental region forΛ maps in a two-to-one way ontC∞ (except at critical points), ande1 = ℘Λ(c1) ande3 = ℘Λ(c3), we can choose a componentW of ℘−1

Λ (U) that containsc1, c2, andc3. We construct a pathα1 from c1 to c3,and a pathα2 from c3 to c2, both contained inW . By symmetry ofF(℘Λ) about criticalpoints, we can construct the paths to lie completely in half of a fundamental domathe lattice; i.e., in = {z ∈ C: z = sλ1 + tλ2, s ∈ [0,1], t ∈ [0, 1

2]}. Moreover we canconstruct the paths to be simple (no self-intersections) and such thatα1 ∩ α2 = {c3} (usinga straightforward topological argument).

For any pathα : [0,1] → C, by −α we denote the path−α(t) = α(1− t), which is justα traversed in reverse order. We extend the pathsα1 andα2 in W by symmetry of eachFatou component about a critical point in such a way that we obtain a pathβ1 from c3throughc1 to c3 − λ2, and a pathβ2 from c3 − λ1 throughc2 to c3. Again by symmetry,these paths extend to a loopq insideW which consists of

β1 ∪ −(β2 − λ2) ∪ −(β1 − λ1) ∪ β2.

Furthermore, since the original paths avoid the pole at 0,q does as well. The loopq isno longer contained in but it is contained completely inW . We claim thatq bounds afundamental region forΛ (the bounded component ofq).

We denote byQ the closed bounded region in the plane with boundaryq. To prove theclaim we observe that by constructionQ + λ1,Q − λ1,Q + λ2, andQ − λ2 are disjointfrom Q except for one boundary curve in each case. By an induction argument wethat the setsQ+mλ1 +nλ2, m,n ∈ Z, tile the plane and are pairwise disjoint except atboundary curves.

Thus q is a loop lying in one component of the Fatou set, and by constructionq issymmetric with respect to 0; that is, ifz ∈ q then−z ∈ q.

Let K denote the unbounded component of the complement ofq. By the symmetry ofq with respect to the origin, we have that 0 is not inK . But thenq is a loop in the Fatou


wt if

e

the

e

el

n

es in

by

t

),

n

set separating 0∈ J (℘Λ) from ∞ ∈ J (℘Λ), soJ (℘Λ) is not connected. It remains to shothat each component ofJ (℘Λ) consists of just one point, which we do by showing thaC is a component ofJ (℘Λ), then diam(C) = 0 (using the Euclidean metric onQ).

We note that the vertices ofQ are the critical pointsc3, c3 − λ2, −c3, andc3 − λ1, thepole (at 0) for℘Λ is in the interior ofQ since the boundary ofQ is in the Fatou set. Wlabel each fundamental region as follows:

Q = Q0,0,

Qm,n = Q0,0 + mλ1 + nλ2, m,n ∈ Z,

and note that any pair of fundamental regions corresponding to distinct(m,n) pairs areeither disjoint or intersect along one boundary segment. FurthermoreC = ⋃

m,n∈ZQm,n.

Since℘Λ is hyperbolic,J (℘Λ) is compact, and we have a double toral band alongboundary ofQ, using Theorem 3.10 and Corollary 3.11 we can finds ∈ N and a neigh-borhoodU of J = J (℘Λ) ∩ Q, with U ⊂ intQ on which|(℘s

Λ)′(z)| � cr with r > 1 forall z ∈ U on which℘s

Λ is defined. By compactness ofJ we can chooseU to be simplyconnected and so that we have a Jordan curveδ = ∂U ⊂ F(℘Λ) such thatJ lies in thebounded component ofδ. By construction ofδ, all critical points and all critical values liin the unbounded component ofδ.

By symmetry, we translate the setsJ andU by the elements of the lattice, and we labthese as follows:

J0,0 = J ; Jm,n = J0,0 + mλ1 + nλ2,

and

U0,0 = U ; Um,n = U0,0 + mλ1 + nλ2.

We also defineδm,n = ∂Um,n = δ + mλ1 + nλ2; this gives a disjoint collection of Jordacurves lying in the Fatou set.

We have that

J (℘Λ) ={z ∈ C∞: ℘k

Λ(z) ∈⋃

m,n∈Z

Um,n for all k for which℘kΛ is analytic

}

and it is clear from the construction that there are no critical points or critical valu⋃m,n∈Z

Um,n. Given any pointz ∈ J (℘Λ) \ {∞}, we have thatz ∈ Jm,n, so δm,n givesa separation ofz from ∞ (sincez lies in the bounded component of its complementconstruction). Therefore{∞} is a component ofJ (℘Λ).

DefiningJ = {z ∈ C: ℘kΛ(z) ∈ ⋃

m,n∈ZUm,n for all k � 1}, we can write the Julia se

as the following disjoint union:

J (℘Λ) = J ∪A∪ {∞};these are the points that stay inC under all iterations of℘Λ, the prepoles (including polesand the point at∞, respectively. For each(i, j) ∈ Z

2, the set℘−1Λ Ui,j consists of two

disjoint sets in the fundamental periodQm,n for every m,n, and℘Λ is a holomorphiccovering map of each component ontoUi,j ; sinceUi,j is simply connected, the restrictio


r

the

-

of ℘Λ to each component is a homeomorphism of that component ontoUi,j . Therefore wecan define branches:

p(i,j)

m,n,0,p(i,j)

m,n,1, m,n, i, j ∈ Z,

of ℘−1Λ on Ui,j , mapping intoQm,n, and we see that the collection of setsp

(i,j)m,n,k(Ui,j ),

m,n, i, j ∈ Z, k = 0,1, are pairwise disjoint compact subsets.We proceed inductively in this way using the following notation. Write

Ui,j (m1, n1, k1) = p(i,j)m1,n1,k1

(Ui,j

).

Then applying another local inverse gives:

Ui,j

((m1, n1, k1), (m2, n2, k2)

) = p(m1,n1)m2,n2,k2

(Ui,j (m1, n1, k1)

),

and so on. LettingΦd denote a triple(md,nd, kd), for each fixedi, j ∈ Z we obtain pair-wise disjoint compact sets of the form:

Ui,j (Φ1,Φ2, . . . ,Φd) = p(i,j)Φd

(Ui,j (Φ1,Φ2, . . . ,Φd−1)

).

By construction, we have

℘−dΛ Ui,j =

⋃Φ1,...,Φd

Ui,j (Φ1,Φ2, . . . ,Φd);

from this it follows that ifU∗ = ⋃i,j∈Z

Ui,j , then

℘−dΛ U∗ =

⋃i,j

⋃Φ1,...,Φd

Ui,j (Φ1,Φ2, . . . ,Φd)

which is still a pairwise disjoint union of compact sets.Furthermore,

J =∞⋂

d=0

℘−dΛ U∗.

Now let V be any component ofJ ; thenV can lie in at most oneUi,j (Φ1,Φ2, . . . ,Φd)

for eachd .By the expansion on the Julia set, we see that

diam(Ui,j (Φ1,Φ2, . . . ,Φd)

)� r−[d/s]diamUi,j

(where[d/s] denotes the integer part ofd/s), which goes to 0 asd → ∞, so the diameteof each component inJ is 0, making each component a single point.

It remains to consider any pointω ∈ A; our first observation is that eachUm,n containsexactly one pole by construction, namelyωm,n = mλ1+nλ2. For any fixed(m,n) ∈ Z×Z,and anyz ∈ J (℘Λ), z �= ωm,n, we show the two points lie in separate components ofJulia set. Ifz /∈ Um,n, then δm,n ⊂ F(℘Λ) separates the two points. Ifz ∈ Um,n, then℘Λ(z) �= ∞, so℘Λ(z) ∈ Ui,j for some(i, j) ∈ Z × Z. We apply a branch of℘−1

Λ to the

curveδi,j ; using the notation above we choose the branch so that the loopγ = p(i,j)m,n,kδi,j

containsz in its bounded component. It then follows thatωm,n is in the unbounded component of the complement ofγ , soz andωm,n are not in the same component ofJ (℘Λ).


e.

ee

nd

ritical

s

have a

st

We proceed inductively onκ . Assume thatω ∈ Aκ \ Aκ−1 andz ∈ J (℘Λ), z �= ω. Bythe inductive hypothesis, we can assume thatz is not an orderj prepole for anyj =1, . . . , κ − 1. Then℘κ

Λ is defined forz andω; we track the sequences ofUi,j containingthe iteratesz,℘Λ(z), . . . ,℘κ

Λ(z) andω,℘Λ(ω), . . . ,℘κΛ(ω) and stop when they disagre

If z ∈ Um,n andω /∈ Um,n, then the curveδm,n separatesz andω. If ℘jΛ(z) ∈ Um,n and

℘jΛ(ω) /∈ Um,n, then the appropriate choice of branch for℘

−jΛ applied toδm,n will yield

a simple closed curve whose complement hasz in the bounded component andω in theunbounded component.

Finally if z,ω and the forward iterates ofz andω share the sameUi,j itinerary underthe firstκ − 1 iterates of℘Λ, then one of two possibilities occurs. Ifz is also an orderκprepole then both points map after exactlyκ − 1 steps to a polezo ∈ Uio,jo ; we can thenfind one of the 2κ−1 preimages ofδio,jo under℘κ−1

Λ , which will have to separatez from ω

as they are distinct. Otherwise℘κΛ(z) ∈ Ua,b, for some(a, b) ∈ Z × Z, so the appropriat

pullback of δa,b under a branch of℘−κΛ will yield the desired separation. Therefore w

have shown that every point inJ (℘Λ) lies in its own component so thatJ (℘Λ) is a Cantorset. �Corollary 3.13. If ℘Λ is hyperbolic and all the critical values of℘Λ are contained in oneFatou componentW , thenW is the basin of attraction for an attracting fixed point aF(℘Λ) = W .

Proof. We apply Theorem 3.12 and Proposition 3.1.�Corollary 3.14. If all three critical values are in the same Fatou componentW of F(℘Λ),then all critical points are inW , W is a double toral band, andJ (℘Λ) has infinitely manycomponents.

Proof. This follows from the proof of Theorem 3.12.�Double toral bands for elliptic functions

Our main result is that the existence of a double toral band that contains all of the cvalues for a hyperbolic elliptic function ensures thatJ (f ) is a Cantor set.

Theorem 3.15. If W is a double toral band forf that contains all of the critical valueandf is hyperbolic, thenF(f ) = W andJ (f ) is a Cantor set.

Proof. We use essentially the same proof as in Theorem 3.12. By hypothesis, wesimple closed loopq contained inF(f ) such thatq bounds a fundamental region forΛ

(the bounded component ofq).We denote byQ the closed bounded region in the plane with boundaryq. Let K denote

the unbounded component of the complement ofq. Since every fundamental region mucontain at least one pole of ordern � 2, we have at least one pole inQ, saypo. Thereforepo ∈ J (f ) is not in K . But thenq is a loop in the Fatou set separatingpo ∈ J (f ) from∞ ∈ J (f ), soJ (f ) is not connected.


y;

ftation

et has

ia set.e

avy) andve thee.

s

We show that each component ofJ (f ) consists of just one point using hyperbolicitas before it is enough to look atQ and show that ifC is a component ofJ (℘Λ) ∩ Q, theneither diam(C) = 0 orC consists of exactly one prepole.

In particular, since all critical values are contained inW we can find a loopδ in Q,bounding a neighborhoodU of J (f ) ∩ Q which lies in the bounded component ofδ,and all of the critical points and critical values lie in the unbounded component ofδ. ByCorollary 2.2, for each(i, j) ∈ Z × Z, the setf −1(Ui,j ) consists ofn disjoint sets in thefundamental periodQm,p for everym,p (wheren is the order off ). The rest of the proofollows exactly as in Theorem 3.12 using Theorem 3.10 and Corollary 3.11. (The nobecomes unmanageable so we omit the details.)

Since the Julia set is totally disconnected, it follows immediately that the Fatou sexactly one component which is therefore completely invariant.�

4. An elliptic function with a Cantor Julia set

In this section we prove the existence of an elliptic function that has a Cantor JulThe elliptic function is of the formfΓ (z) = (℘Γ (z) + 1)/℘Γ (z) with a real square latticΓ = [γ, γ i]; Γ is the period lattice for both℘Γ andfΓ . Properties of℘Γ are used tocontrol the behavior offΓ ; in this way we construct a hyperbolic elliptic function withdouble toral band. The Julia set offΓ is contained in the set of dark points (colored nablue) in Fig. 8; we also show the boundary of a period parallelogram (in medium bluethe attracting fixed point (in orange). The basic idea behind the construction is to mopoles off the real and imaginary axes just by the definition offΓ , and then to choose thperiod latticeΓ to create an attracting fixed point with the axes in the attracting basin

For our starting point, we define the latticeΛ = [λ, iλ], λ ∈ R, λ > 0, to be the latticeassociated with the invariantsg2 = 4, g3 = 0. By Proposition 4.1 below the critical value

Fig. 8. A Cantor Julia set.


he

is a

helyly

half

d

of ℘Λ are 0,1,−1; let e1,Λ = 1. Using the tables in [14],λ ≈ 2.62. This lattice is referredto as thestandard square lattice. The first result holds for any square lattice.

Proposition 4.1. If Γ is a square lattice theng3 = 0; in this case, the critical values of℘Γ

are e3 = 0, e1 = √g2/2, ande2 = −e1.

Given any square latticeΓ and the elliptic functionfΓ (z) = (℘Γ (z) + 1)/℘Γ (z), weprove some basic properties about the function.

Lemma 4.1. LetΓ be a square lattice. Then the points(γ +γ i)/2+Γ are the poles offΓ .The critical points offΓ are the lattice pointsΓ , and the pointsγ /2+ Γ andγ i/2+ Γ .

Proof. Since℘Γ (z) = 0 if and only ifz = (γ +γ i)/2+Γ by Proposition 4.1(γ +γ i)/2+Γ are the poles offΓ . Sincef ′

Γ (z) = −℘′Γ (z)/(℘Γ (z))2, thenf ′

Γ (γ /2 + Γ ) = 0 andf ′

Γ (γ i/2+ Γ ) = 0 since℘′Γ is zero at those points.

To see that the poles of℘Γ are critical points offΓ , we observe using Eq. (1)

f ′(0) = limx→0

f ′(x) = limx→0

℘′Γ (x)

(℘Γ (x))2

= limx→0

√4(℘Γ (x))3 − g2℘Γ (x)

(℘Γ (x))2= 0. �

The next lemma was proved in [10] for℘Γ , but the proof remains the same for telliptic functionfΓ .

Lemma 4.2. If Γ is any lattice andfΓ (z) = (℘Γ (z) + 1)/℘Γ (z) then

(1) fΓ (z) = fΓ (−z) for all z (f is even).(2) (−1)J (fΓ ) = J (fΓ ) and(−1)F (fΓ ) = F(fΓ ).(3) If Γ is real thenJ (fΓ ) = J (fΓ ) andF(fΓ ) = F(fΓ ).

Remark. Parts (1) and (2) of Lemma 4.2 are true for any elliptic function whichrational function of℘Γ for any latticeΓ . If Γ is a real lattice then (3) holds similarly.

For any real square latticeΓ = [γ, γ i], ℘Γ is real on the small square formed by thalf periods ofΓ (see [7]). Along the real interval[0, γ /2], ℘Λ decreases monotonicalfrom ∞ to e1. Moving vertically fromγ /2 to (γ + γ i)/2, ℘Γ decreases monotonicalfrom e1 to 0. As we move along the other two legs of the small square formed by theperiods,℘Γ decreases from 0 toe2 = −e1 to −∞.

Our example is constructed using the lattice defined in the following lemma.

Lemma 4.3. We defineΓ to be the square lattice associated with the invariantsg2(Γ ) =16, g3(Γ ) = 0. ThenΓ = kΛ, wherek = 4

√1/4 andΛ is the standard square lattice, an

℘Γ (γ /2) = 2. Therefore the critical values of℘Γ are 2,−2, and0.


g

Proof. SinceΓ andΛ are both square, there exists ak such thatΓ = kΛ. By Lemma 2.4

k4 = g2(Λ)

g2(Γ )= 1

4.

By Eq. (2)

℘Γ

(γ

2

)= ℘kΛ

(kλ

2

)= 1

k2℘Λ

(λ

2

)= 1

k2= 2. �

For the rest of this section,Γ refers to the lattice defined in Lemma 4.3.

Lemma 4.4. For all z ∈ R, we have0� |f ′Γ (z)| < 1.

Proof. SinceΓ is a real lattice,fΓ is a real function, sof ′Γ (z) is real ifz is. For simplicity

of notation, we usef and℘ (and suppress the latticeΓ which is fixed throughout). UsinEq. (1), it is enough to show that

(f ′)2 = (

4℘3 − 16℘)/℘4 = 4

℘− 16

℘3< 1

on the periodic interval[0, γ ].Moreover, by symmetry about the critical points, we need only look at[0, γ /2]. By

Lemma 4.1,f ′(z) = 0 at the endpoints of this interval and the mapf is strictly increasingon (0, γ /2), since℘ decreases from∞ to 2 there. Sincef is not constant,f ′ > 0 and hasa maximum value occurring at a pointc ∈ (0, γ /2); the maximum forg = (f ′)2 occurs atc as well so we solve forc

g′(z) = 4℘′(

12− ℘2

℘4

),

which is zero in(0, γ /2) when℘(z) = 2√

3. This occurs at precisely one pointc, and atthat point we have that:

(f ′)2

(c) = 4

2√

3− 16

24√

3= 4

3√

3< 1. �

Theorem 4.5. fΓ is hyperbolic and has a double toral Fatou band.

Proof. We know that℘Γ is real on the small square formed by the half periods ofΓ , andthusfΓ is real there are well. We note first thatfΓ is strictly increasing on[0, γ /2] since℘Γ is strictly decreasing there. SincefΓ (0) = 1 andfΓ (γ /2) = 3/2, fΓ increases from1 to 3/2 on the interval[0, γ /2]. By Lemma 4.2 (1) and periodicity offΓ we have thatfΓ :R → [1,3/2].

Clearly there is at least one real fixed point. It cannot occur on the interval[0, γ /2]sinceγ /2 ≈ 2.62/(2

√2) < 1. Similarly, there are no fixed points on[γ,∞) sinceγ ≈

2.62/√

2 > 3/2. SincefΓ is even, we have thatfΓ is strictly decreasing from 3/2 to 1 onthe interval[γ /2, γ ]. Thus there can be only one fixed point on[γ /2, γ ]. This fixed pointis attracting using Lemma 4.4.


real

s 2.10

.1, all

Juliampless of

a su-quare

les. Int with

d Juliaut aree Julia

Thus all points inR are attracted to this real fixed point. This implies that the entireaxis is contained in the Fatou set. Since℘Γ is real on the vertical linel from 0 toγ i/2,we have thatfΓ is real there as well. The invariance of the Fatou set implies thatl liesin the Fatou set, and thus the entire imaginary axis lies in the Fatou set by Lemmaand 4.2(1). By Lemma 2.10 we have that the square with vertices 0, γ, γ + γ i, γ i lies in acomponentW of the Fatou set, hence we have a double toral band. Using Lemma 4critical points lie inW andf is hyperbolic. �Theorem 4.6. Let Γ be the square lattice associated with the invariantsg2(Γ ) = 16,g3(Γ ) = 0. ThenJ (fΓ ) is a Cantor set.

Proof. We apply Theorems 4.5 and 3.15.�

5. Special lattices and connectivity

We return in this section to the study of Julia sets of the Weierstrass℘ function. Weshowed in Corollary 3.3 of Section 3 that triangular lattices give rise to connectedsets for℘. Here we collect results on other special lattice shapes and give some exaof Julia sets which contain the entire real and imaginary axes. We also give exampleΛ

such thatJ (℘Λ) is connected but the intersection ofJ (℘Λ) with R (and other lines) is aCantor set.

In [9] we proved that every Weierstrass elliptic function on a square lattice withperattracting fixed point is connected, and in [9,10] we constructed examples of slattices with this property. Lemma 2.6 provides a formula for obtaining these exampSection 5.3, we prove that we obtain Cantor sets for the intersection of the Julia sethe axes, lattice boundaries, and lattice diagonals.

5.1. Real rectangular lattice

We prove the existence of a real rectangular (nonsquare) lattice with a connecteset. In this example, two critical values lie in different components of the Fatou set bassociated to the same superattracting fixed point, and one critical value lies in thset.

Theorem 5.1. Let a = 3, and letΓ = [γ1, γ2], γ1 > 0 be the real rectangular lattice withinvariantsg2(Γ ) = 4(a2+a +1)/a2 andg3(Γ ) = −4(a +1)/a2. If k = 3

√2/(3γ1) (taking

the real root) andΛ = kΓ then the Julia set of℘Λ is connected.

Proof. Using Eq. (4) we have that

e1,Γ = 1, e2,Γ = −1− a

a= −4

3, e3,Γ = 1

a= 1

3.

Using Eq. (2) we have

e1,Λ = ℘Λ

(λ1

)= ℘kΓ

(kγ1

)= 1

2℘Γ

(γ1

)= 3λ1

,
2 2 k 2 2


nt,

).

oring

ore

ttices.

ts for

t

nrealsame

atpoint,lre can

are

and thuse1,Λ = 3λ1/2 is a superattracting fixed point. A similar argument givese3,Λ =λ1/2, and thus℘Λ(e3,Λ) = e1,Λ.

We claim thate1 and e3 cannot lie in the same Fatou component. LetU denote theFatou component containinge1. SinceU is a superattracting invariant Fatou componewe have that there is a conformal change of coordinates fromU onto the open unit diskDwhich conjugates℘Λ|U to thez → z2 power map onD (cf. [18, Section 68, Theorem 4]If e3,Λ ∈ U then the local coordinate change extends throughout a regionV ⊂ U withe3,Λ ∈ ∂V . However, the local coordinate change cannot extend over to a neighbperiod parallelogram since℘Λ is periodic but the power map is not. Thereforee3,Λ /∈ U .

Using Eq. (4), we havee2,Λ = −2λ1 ∈ J (℘Λ). Thus no Fatou component contains mthan one critical value andJ (℘Λ) is connected by Theorem 3.1.�5.2. Real rhombic lattices

We turn to a discussion of some possibilities for Fatou sets for rhombic period laThe lattices in this section are all real rhombic lattices and have the formΛ = [λ1, λ1]. Webegin with two propositions about the critical orbits and possible Fatou componenreal rhombic lattices.

Proposition 5.1. If Λ is a real rhombic lattice then

(1) There is one real critical valuee3; e3 < 0 if g3 < 0 ande3 > 0 if g3 > 0.(2) There are two complex critical values satisfyinge1 = e2.

Proof. In [9] we showed thate3 has the same sign asg3. Theorem 2.5(1) implies thae1 = e2. �Proposition 5.2. For any real rhombic latticeΛ one of the following must occur:

(1) J (℘Λ) = C∞;(2) There exist one real postcritical orbit and two conjugate postcritical orbits; therefore

there are at most two different types of periodic Fatou components. If the nocritical values lie in the Fatou set, then they are associated with cycles with theperiod and multiplier.

Proof. Suppose the Fatou set in nonempty. SinceΛ is real, Theorem 2.5(1) implies th℘Λ maps the real line to the real line, and thus the postcritical orbit of the real criticalnever moves off of the real axis. By Proposition 5.1e1 = e2. Applying Theorem 2.5(1)we have that℘n(e1) = ℘n(e2) = ℘n(e2) for all n � 0. Since℘′

Λ(z) = ℘′Λ(z), the nonrea

critical points are associated with cycles of the same period and multiplier. Thus thebe at most two different types of Fatou components.�Theorem 5.2. If Λ is a real rhombic lattice such that the complex critical valuesassociated with nonrepelling complex cycles and the real critical value is inJ (℘Λ) thenthe real and imaginary axes are contained inJ (℘Λ).


e,

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f

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-he, the

n) it-zoomvalues con-

Proof. SinceΛ is real, Theorem 2.5(1) implies that℘Λ maps the real line to the real linand thus no point onR can leave the real axis.

Since the real critical value is inJ (℘Λ), then the real postcritical orbit is either assoated with a Siegel disk or is not associated to any Fatou cycle by Theorems 2.7 andinterval in R can lie within a Siegel disk component because℘Λ :R → R, which wouldcontradict that℘n

Λ is conjugate to an irrational rotation of the unit disk.If the complex cycle is attracting or parabolic then noz ∈ R can approach the comple

cycle because it cannot leave the real axis. If the complex cycle is associated with adisk, then again no point on the real axis can lie within the cycle.

Therefore, the entire real axis must lie inJ (℘Λ). SinceΛ is real, we have that℘Λ(iy) ∈R for y ∈ R (see [9]), and thus the imaginary axis must lie inJ (℘Λ) by the invariance othe Julia set. �

We next give some experimental examples of the Julia and Fatou sets of Weieelliptic functions on real rhombic period lattices.

Example 5.3. A real rhombic lattice with a connected Julia set.

If g2 ≈ 14.4676 andg3 ≈ −13.0598 thenΛ is real rhombic. Here,℘Λ has a superattracting real fixed point atp ≈ −2.25, which is plotted (in orange) in Fig. 9, and tlightest points (colored yellow) in Figs. 9 and 10 iterate to this fixed point. Furthercomplex critical values head to an attracting two-cycle at≈ {1.2554+0.428441i,1.2554−0.428441i}, also plotted (in orange) in Fig. 9; the medium shaded points (colored taerate to this cycle. The Julia set is the set of dark points (navy blue). In Fig. 10, wein on the component containing the superattracting fixed point. Since each criticalis contained in a distinct Fatou component, Theorem 3.1 implies that the Julia set inected.

Fig. 9. Fixed points and the Fatou set of Example 5.3.


agi-

es

redrate2 is

Fig. 10. Superattracting fixed point of Example 5.3.

Fig. 11. Fixed points of Example 5.4.

Example 5.4. A real rhombic lattice with a nonempty Fatou set where the real and imnary axes are inJ (℘Λ).

If g2 ≈ 0.7 andg3 ≈ 7.15124 thenΛ is real rhombic. The real critical pointλ3/2 =(λ1 + λ1)/2 satisfies℘4

Λ(λ3/2) = 3λ3 ∈ J (℘Λ), and hence the real and imaginary axlie in the Julia set by Theorem 5.2. There are attracting fixed points atp ≈ −0.652+0.964i andp ≈ −0.652−0.964i, marked (in orange) in Fig. 11; the lightest points (coloyellow) iterate to the fixed point atp, and the medium shaded points (colored tan) iteto the fixed point atp. The Julia set is the set of dark points (colored navy blue). Fig. 1a zoom in to the real critical pointλ3/2≈ 1.088.


tuenceylatticehat

for thee can

Fig. 12. Critical point of Example 5.4.

Fig. 13.J (℘Λ) for aΓ with a superattracting fixed point.

5.3. Cantor intersections

We show first that for any real rectangular square latticeΓ with an attracting fixed poin(for ℘Γ ), the intersection of the Julia set with the real axis is a Cantor set. As a conseqwe obtain Cantor intersections ofJ (℘Γ ) with other lines through the origin as well. Bsymmetry the result holds for certain lines parallel to these. We fix a real squareΓ = [γ, γ i], γ > 0 for which there is an attracting fixed point. It was shown in [9] tthere are infinitely many distinct values ofg2 > 0, or equivalentlyγ , for which℘Γ has asuperattracting fixed point (see Fig. 1 for an example). Fig. 13 shows the Julia setlattice in Fig. 1 restricted to one period parallelogram centered at the origin. The latticbe generated byγ ≈ 2.395, and the superattracting fixed point occurs ate1 = γ /2≈ 1.197.


y inints] and

r

n-

n-

e,

we

By Eq. (1) it is clear that both the fixed point and its derivative vary analyticallg2 > 0 (andγ ), and therefore nearby ing2 parameter space there are attracting fixed poas well. The next two propositions are an amalgamation of known results from [9[10].

We label the periodic intervals for℘Γ |R by

Ik = [kγ, (k + 1)γ

], k ∈ Z;

the endpoints ofIk are the poles of℘Γ on R.We describe the possibilities forJ (℘Γ ).

Proposition 5.3. For any real rectangular square latticeΓ one of the following must occu:

(1) J (℘Γ ) = C∞;(2) There exists exactly one (super)attracting cycle for℘Γ , and the immediate basin co

tainse1; this is the only periodic cycle forF(℘Γ );(3) There exists exactly one rationally neutral cycle for℘Γ , and the immediate basin co

tainse1; this is the only periodic cycle forF(℘Γ );(4) In the case of(2) or (3), the following also hold:

(a) The nonrepelling periodic orbit lies completely in[e1,∞).

(b) Supposekoλ � e1 < (ko +1)λ; if A = {p1, . . . , pn} denotes the nonrepelling cyclthenA ∩ Iko �= ∅.

We add to this in the case of an attracting fixed point.

Proposition 5.4. Assume thatΓ is a real rectangular square lattice such thatto is anattracting fixed point. Thento ∈ R, and:

(1) We have real critical valuese1 > 0, e2 = −e1 , ande3 = 0.(2) ℘Γ :R → R ∪ ∞; moreover℘Γ ([e1,∞]) = [e1,∞].(3) If Iko is the interval containinge1 we let cko denote the critical point inIko . Then

to ∈ Iko and there is a repelling fixed pointp ∈ Iko such that, writingp = cko + q,0< q < γ/2, the real immediate basin of attraction forto is B = (cko − q, cko + q);

(4) ℘Γ (B) = B.

In the next proposition we prove the existence of Cantor intersections forJ (℘Γ ); wecarry over all of the notation from above.

Proposition 5.5. If Γ is a real rectangular square lattice with an attracting fixed point,have the following:

(1) J (℘Γ ) ∩ R is a Cantor set;(2) J (℘Γ ) ∩ {z: z = iy, y ∈ R} is a Cantor set;(3) J (℘Γ ) ∩ {z: z = a + ia, a ∈ R} is a Cantor set;(4) J (℘Γ ) ∩ {z: z = a − ia, a ∈ R} is a Cantor set.


eriodic

the;

e

he

lyal

ofnefor

hic

ious

we

Proof. Because of the symmetry of the Julia and Fatou sets with respect to each pinterval, it follows from Propositions 5.3 and 5.4 that℘Γ (Ij ) = [e1,∞] for all j , and that℘−1

Γ (B) consists of infinitely many disjoint open intervals, one in eachIj . We label thoseintervalsBj ⊂ Ij ; proceeding as in [5, Proposition 2 for the tangent family], we labelintervals complementary to theBj ’s usingCj such thatCo contains the pole at the originthenCko+1 contains the repelling fixed pointp.

It follows from our construction and the strict monotonicity of℘′Γ for rectangular squar

lattices that℘Γ :Cj → [e1,∞]\(B ∩[e1,∞]) for eachj and|℘′Γ (x)| > 1 for eachx ∈ Cj .

Defining

JR ={x ∈ R | ℘n

Γ (x) ∈⋃

j�ko

Cj for all n for which℘nΓ is analytic

},

we have that

z ∈ JR ⇐⇒ z ∈ J (℘Γ ) ∩ R.

We show thatJR is a Cantor set by showing that diam(℘−nΓ Cj ) → 0 asn → ∞, and that

each real prepole is a component ofJR as in the proof of Theorem 3.12 and following tmethod outlined in [5].

For any pointx ∈ (−∞,−e1), ℘−1Γ (x) is purely imaginary and consists of infinite

many pairs of points of the form±ia, a ∈ R. More precisely, in each imaginary interviIk , k ∈ Z, there are exactly two preimages. Therefore for eachk, the set

℘−1Γ (JR) ∩ [

i kγ, i (kγ + γ /2)] = J (℘Γ ) ∩ [

i kγ, i (kγ + γ /2)]

is a homeomorphic image ofJR and hence a Cantor set. If we denote byJRi the setJ (℘Γ ) ∩ {z = iy}, then we have shown thatJRi is a Cantor set.

We now consider the set℘−1Γ (JRi). First, we know that the preimage of 0 consists

all critical points of the form12(kγ + i kγ ); i.e., 0 is a critical value so there is only o

preimage in each period parallelogram forΓ . We define the central diagonal intervalseachk ∈ Z: Dk = {x + i x: x ∈ Ik} anddk = {x − i x: x ∈ Ik}. If w ∈ JRi is of the formw = iy, y < 0, then we have two preimages ofw in eachDk ; if w = iy, y > 0, then wehave two preimages in eachdk . As above, taking the countable union of homeomorpcopies of the Cantor setJRi gives the result. �

By symmetry ofJ (℘Γ ) with respect to the lattice, we do not need to restrict the prevresult to lines through the origin. Ifγ ∈ Γ , andA is a set inC, then

A + γ = {z + γ, z ∈ A}.

Corollary 5.5. If Γ is a real rectangular square lattice with an attracting fixed point,have the following for anyγ ∈ Γ :

(1) J (℘Γ ) ∩ (R + γ ) is a Cantor set;(2) J (℘Γ ) ∩ ({z: z = iy, y ∈ R} + γ ) is a Cantor set;(3) J (℘Γ ) ∩ ({z: z = a + ia, a ∈ R} + γ ) is a Cantor set;(4) J (℘Γ ) ∩ ({z: z = a − ia, a ∈ R} + γ ) is a Cantor set.


for

he

l dis-g theplace.exam-roved

s 11

s 11

h. 22

atics,

. Sci.

ourier

natsh.

yn. 8

niver-

ath. 24

, Bull.

Corollary 3.3 states that every triangular latticeΓ gives rise to a connected Julia set℘Γ ; however the following results also holds.

Proposition 5.6. If Γ is a real triangular lattice with an attracting fixed point, we have tfollowing for anyγ ∈ Γ :

(1) J (℘Γ ) ∩ (R + γ ) is a Cantor set;(2) J (℘Γ ) ∩ (e2πi/3

R + γ ) is a Cantor set;(3) J (℘Γ ) ∩ (e4πi/3

R + γ ) is a Cantor set.

Acknowledgements

The first author thanks J. Kotus, M. Urbanski, L. Keen, and R. Devaney for usefucussions related to this work, and especially A. Clark and M. Urbanski for hostinconference at the University of North Texas at which many of the discussions tookThe second author thanks M. Moreno Rocha for asking the question answered by theple in Section 5.3. The referee is also acknowledged for helpful comments that impthe paper.

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[3] I.N. Baker, J. Kotus, Y. Lü, Iterates of meromorphic functions IV: Critically finite functions, Results Mat(1992) 651–656.

[4] W. Bergweiler, Iteration of meromorphic functions, Bull. Amer. Math. Soc. 29 (2) (1993) 151–188.[5] R. Devaney, L. Keen, Dynamics of tangent, in: Dynamical Systems, in: Lecture Notes in Mathem

vol. 1342, Springer, Berlin, 1988, pp. 105–111.[6] R. Devaney, L. Keen, Dynamics of meromorphic maps with polynomial Schwarzian derivative, Ann

École Norm. Sup. (4) 22 (1989) 55–81.[7] P. DuVal, Elliptic Functions and Elliptic Curves, Cambridge University Press, Cambridge, 1973.[8] A.E. Eremenko, M.Y. Lyubich, Dynamical properties of some classes of entire functions, Ann. Inst. F

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(1999) 133–145.[13] J. Kotus, M. Urbanski, Hausdorff dimension and Hausdorff measures of Julia sets of elliptic functions

London Math. Soc. 35 (2003) 269–275.[14] L. Milne-Thomson, Jacobian Elliptic Function Tables, Dover, New York, 1950.[15] J. Milnor, On rational maps with two critical points, Experimental Math. 9 (2000) 480–522.


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Connectivity properties of Julia sets of Weierstrass elliptic functions